Dirk Kreimer

Hopf Algebras, Renormalization and Noncommutative Geometry

Abstract:We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated to the NCG computations of tranverse index theory for foliations.

Renormalization in Quantum Field Theory and the Riemann-Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem

Abstract:This paper gives a complete selfcontained proof of our result announced in [6] showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann–Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra which is commutative as an algebra. It is the dual Hopf algebra of the enveloping algebra of a Lie algebra whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group G is the group of characters of . We shall then show that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop where C is a small circle of complex dimensions around the integer dimension D of space-time. Our main result is that the renormalized theory is just the evaluation at z=D of the holomorphic part γ+ of the Birkhoff decomposition of γ. We begin to analyse the group G and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. The analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title.

Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group

Abstract: We showed in Part I that the Hopf algebra ℋ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group G and that the renormalized theory is obtained from the unrenormalized one by evaluating at ɛ= 0 the holomorphic part γ+(ɛ) of the Riemann–Hilbert decomposition γ−(ɛ)− 1γ+(ɛ) of the loop γ(ɛ)∈G provided by dimensional regularization. We show in this paper that the group G acts naturally on the complex space X of dimensionless coupling constants of the theory. More precisely, the formula g0=gZ1Z3−3/2 for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra ℋ. This allows first of all to read off directly, without using the group G, the bare coupling constant and the renormalized one from the Riemann–Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter ɛ. It also allows to lift both the renormalization group and the β-function as the asymptotic scaling in the group G. This exploits the full power of the Riemann–Hilbert decomposition together with the invariance of γ−(ɛ) under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group G for the full higher pole structure of minimal subtracted counterterms in terms of the residue.

Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops

Abstract It is found that the number, M n , of irreducible multiple zeta values (MZVs) of weight n , is generated by 1 − x 2 − x 3 = Π n (1 − x n ) M n . For 9 ≥ n ≥ 3, M n enumerates positive knots with n crossings. Positive knots to which field theory assigns knot-numbers that are not MZVs first appear at 10 crossings. We identify all the positive knots, up to 15 crossings, that are in correspondence with irreducible MZVs, by virtue of the connection between knots and numbers realized by Feynman diagrams with up to 9 loops.

On Motives associated to graph polynomials

The appearance of multiple zeta values in anomalous dimensions and β-functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions.

Renormalization in quantum field theory and the Riemann-Hilbert problem

We show that renormalization in quantum field theory is a special instance of a general mathematical procedure of multiplicative extraction of finite values based on the Riemann--Hilbert problem. Given a loop γ(z), | z | = 1 of elements of a complex Lie group G the general procedure is given by evaluation of γ+(z) at z = 0 after performing the Birkhoff decomposition γ(z) = γ−(z)−1γ+(z) where γ±(z)G are loops holomorphic in the inner and outer domains of the Riemann sphere (with γ−(∞) = 1). We show that, using dimensional regularization, the bare data in quantum field theory delivers a loop (where z is now the deviation from 4 of the complex dimension) of elements of the decorated Butcher group (obtained using the Milnor-Moore theorem from the Kreimer Hopf algebra of renormalization) and that the above general procedure delivers the renormalized physical theory in the minimal substraction scheme.

Spitzer's Identity and the Algebraic Birkhoff Decomposition in pQFT

In this paper we continue to explore the notion of Rota–Baxter algebras in the context of the Hopf algebraic approach to renormalization theory in perturbative quantum field theory. We show in very simple algebraic terms that the solutions of the recursively defined formulae for the Birkhoff factorization of regularized Hopf algebra characters, i.e. Feynman rules, naturally give a non-commutative generalization of the well-known Spitzers identity. The underlying abstract algebraic structure is analysed in terms of complete filtered Rota–Baxter algebras.

Integrable Renormalization II: The General Case

Abstract.We extend the results we obtained in an earlier work [1]. The cocommutative case of ladders is generalized to a full Hopf algebra of (decorated) rooted trees. For Hopf algebra characters with target space of Rota-Baxter type, the Birkhoff decomposition of renormalization theory is derived by using the double Rota-Baxter construction, respectively Atkinson’s theorem. We also outline the extension to the Hopf algebra of Feynman graphs via decorated rooted trees.

Exact solutions of Dyson–Schwinger equations for iterated one-loop integrals and propagator-coupling duality

Abstract The Hopf algebra of undecorated rooted trees has tamed the combinatorics of perturbative contributions, to anomalous dimensions in Yukawa theory and scalar φ3 theory, from all nestings and chainings of a primitive self-energy subdivergence. Here we formulate the nonperturbative problems which these resummations approximate. For Yukawa theory, at spacetime dimension d=4, we obtain an integrodifferential Dyson–Schwinger equation and solve it parametrically in terms of the complementary error function. For the scalar theory, at d=6, the nonperturbative problem is more severe; we transform it to a nonlinear fourth-order differential equation. After intensive use of symbolic computation we find an algorithm that extends both perturbation series to 500 loops in 7 minutes. Finally, we establish the propagator–coupling duality underlying these achievements making use of the Hopf structure of Feynman diagrams.The Hopf algebra of undecorated rooted trees has tamed the combinatorics of perturbative contributions, to anomalous dimensions in Yukawa theory and scalar

On Overlapping Divergences

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